Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
214 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C7 | D16 |
| commutator | C16 | C22×C4×C7 |
| Frattini, cosocle | C2×C16 | C23×C7 |
| socle | C22×C7 | C2×D16 |
| prime radical | C2×C7×D32 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C7 ◁ C42×C7 ◁ C4×C7×C8 ◁ C4×C7×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C7×D16 | C2 | semidirect |
| 3. C7×#1423 | C2 | semidirect |
| 4. C7×#1415 | C2 | semidirect |
| 5. C4×C7×C32 | C2 | semidirect |
| 8. C7×D32 | C4 | direct (N × H) |
| 14. C4×D32 | C7 | direct (N × H) |
| 18. C7×D16 | C2×C4 | semidirect |
| 25. C7×C32 | C2×C4 | semidirect |
| 27. C4×D16 | C2×C7 | semidirect |
| 28. #1423 | C2×C7 | semidirect |
| 29. #1415 | C2×C7 | semidirect |
| 30. C4×C32 | C2×C7 | semidirect |
| 45. D32 | C4×C7 | direct (N × H) |
| 65. D16 | C2×C4×C7 | semidirect |
| 72. C32 | C2×C4×C7 | semidirect |
| 79. C4×C7 | D32 | direct (N × H) |
| 109. C7 | C4×D32 | direct (N × H) |
| 116. C4 | C7×D32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×D32 | 2 |
| C4×C7×D16 | 2 |
| C7×#1423 | 2 |
| C7×#1415 | 2 |
| C4×C7×C32 | 2 |
| C4×D32 | 7 |